I started this blog about a year ago briefly recommending Rob Lazarsfeld’s book Positivity in Algebraic Geometry, which gives bite-size treatments of many topics hard to find elsewhere.

I’d like to make a stronger case now because it’s an important book. People often give me credit for knowing a lot just because I know what’s in it. It’s rarely on my shelves because it’s almost always in a stack near where I’m working. When I lost my copies in transit between MSRI and Cambridge, I replaced them immediately.*

The title might sound, on the face of it, like something specialized or technical. In fact, positivity is arguably the fundamental difference between algebraic geometry and topology. For example, the intersection multiplicity of two distinct complex curves which meet at a point in a complex algebraic surface S is always positive. As a result, if you know the homology classes of the two curves, then you know the total intersection number N from the cohomology ring of S, and that implies that the “physical” number of intersection points is at most N. This is completely false in topology: you can push around one submanifold to meet another submanifold in as many points as you like. The result is that just knowing the homology class of an algebraic curve controls its geometric properties (it can’t wiggle too much). Much of algebraic geometry builds on this kind of rigidity.

*Not at all painful because Lazarsfeld insisted Springer publish in paperback and keep the price down. Losing Kollár’s Rational Curves on Algebraic Varieties, on the other hand…

2 responses to “Why you should care about positivity”

Let me describe one (rather specialized) example that has come my way. Fix G > P > B a finite-dimensional complex reductive Lie group (say the complexification of a compact group K), a parabolic, and a Borel. Then there is a map G/B -> G/P, and the preimage of a positive cycle on G/P is again positive (of course) on G/B.

Infinite-dimensionally, instead of G/B and G/P aka K/T and K/L, one might have LK/T and LK/K, where L is free loops. So of course we again have LK/T -> LK/K. These spaces LK/T, LK/K can (roughly) again be made algebraic varieties. (More precisely, ind-schemes include into them inducing homotopy equivalences, but whatever.) This map is then algebraic.

But now there is a crazy alternative: identify LK/K with Omega K, based loops. Then there is a map LK/K = Omega K -> LK -> LK/T going the wrong direction. And as a sign that it’s pure topology and not (algebraic) geometry, the pullback of a Schubert cycle on LK/T is not naturally positive on LK/K.

(Thomas Lam came up with these, and calls them affine Stanley symmetric functions, at least in the K = U(n) case.)